Abstract. We prove a conjectured upper bound for the Castelnuovo-Mumford regularity of binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that reg(J G ) ≤ n − 1 for any graph G with n vertices, which is not a path. Moreover, we study the behavior of the regularity of binomial edge ideals under the join product of graphs.
Communicated by J. Walker MSC: 05C40 05C45 16P10 16P40a b s t r a c t Let R be a commutative ring. The total graph of R, denoted by T (Γ (R)) is a graph with all elements of R as vertices, and two distinct vertices x, y ∈ R, are adjacent if and only if x + y ∈ Z (R), where Z (R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ (R)), be the induced subgraph of T (Γ (R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z (R) is not an ideal. In this paper we show that if T (Γ (R)) is a connected graph, then diam(Reg(Γ (R))) diam(T (Γ (R))). Also, we prove that if R is a finite ring, then T (Γ (R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.
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